# catanhf, catanh, catanhl

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< c‎ | numeric‎ | complex

Defined in header ``<complex.h>`` |  |
float complex catanhf( float complex z ); | (1) | (since C99)
double complex catanh( double complex z ); | (2) | (since C99)
long double complex catanhl( long double complex z ); | (3) | (since C99)
Defined in header ``<tgmath.h>`` |  |
#define atanh( z ) | (4) | (since C99)

1-3) Computes the complex arc hyperbolic tangent of `z` with branch cuts outside the interval [−1; +1] along the real axis.
4) Type-generic macro: If `z` has type long double complex, `catanhl` is called. if `z` has type double complex, `catanh` is called, if `z` has type float complex, `catanhf` is called. If `z` is real or integer, then the macro invokes the corresponding real function (atanhf, atanh, atanhl). If `z` is imaginary, then the macro invokes the corresponding real version of atan, implementing the formula atanh(iy) = i atan(y), and the return type is imaginary.

### Parameters

z | - | complex argument

### Return value

If no errors occur, the complex arc hyperbolic tangent of `z` is returned, in the range of a half-strip mathematically unbounded along the real axis and in the interval [−iπ/2; +iπ/2] along the imaginary axis.

### Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

- catanh(conj(z)) == conj(catanh(z))
- catanh(-z) == -catanh(z)
- If `z` is `+0+0i`, the result is `+0+0i`
- If `z` is `+0+NaNi`, the result is `+0+NaNi`
- If `z` is `+1+0i`, the result is `+∞+0i` and FE_DIVBYZERO is raised
- If `z` is `x+∞i` (for any finite positive x), the result is `+0+iπ/2`
- If `z` is `x+NaNi` (for any finite nonzero x), the result is `NaN+NaNi` and FE_INVALID may be raised
- If `z` is `+∞+yi` (for any finite positive y), the result is `+0+iπ/2`
- If `z` is `+∞+∞i`, the result is `+0+iπ/2`
- If `z` is `+∞+NaNi`, the result is `+0+NaNi`
- If `z` is `NaN+yi` (for any finite y), the result is `NaN+NaNi` and FE_INVALID may be raised
- If `z` is `NaN+∞i`, the result is `±0+iπ/2` (the sign of the real part is unspecified)
- If `z` is `NaN+NaNi`, the result is `NaN+NaNi`

### Notes

Although the C standard names this function "complex arc hyperbolic tangent", the inverse functions of the hyperbolic functions are the area functions. Their argument is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse hyperbolic tangent", and, less common, "complex area hyperbolic tangent".

Inverse hyperbolic tangent is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventionally placed at the line segmentd (-∞,-1] and [+1,+∞) of the real axis.

The mathematical definition of the principal value of the inverse hyperbolic tangent is atanh z =

ln(1+z)-ln(z-1)
2

.

For any z, atanh(z) =

atan(iz)
i

### Example

Run this code

    #include <stdio.h>
    #include <complex.h>

    int main(void)
    {
        double complex z = catanh(2);
        printf("catanh(+2+0i) = %f%+fi\n", creal(z), cimag(z));

        double complex z2 = catanh(conj(2)); // or catanh(CMPLX(2, -0.0)) in C11
        printf("catanh(+2-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));

        // for any z, atanh(z) = atan(iz)/i
        double complex z3 = catanh(1+2*I);
        printf("catanh(1+2i) = %f%+fi\n", creal(z3), cimag(z3));
        double complex z4 = catan((1+2*I)*I)/I;
        printf("catan(i * (1+2i))/i = %f%+fi\n", creal(z4), cimag(z4));
    }

Output:

    catanh(+2+0i) = 0.549306+1.570796i
    catanh(+2-0i) (the other side of the cut) = 0.549306-1.570796i
    catanh(1+2i) = 0.173287+1.178097i
    catan(i * (1+2i))/i = 0.173287+1.178097i

### References

- C11 standard (ISO/IEC 9899:2011):

  - 7.3.6.3 The catanh functions (p: 193)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - G.6.2.3 The catanh functions (p: 540-541)

  - G.7 Type-generic math <tgmath.h> (p: 545)

- C99 standard (ISO/IEC 9899:1999):

  - 7.3.6.3 The catanh functions (p: 175)

  - 7.22 Type-generic math <tgmath.h> (p: 335-337)

  - G.6.2.3 The catanh functions (p: 475-476)

  - G.7 Type-generic math <tgmath.h> (p: 480)

### See also

casinhcasinhfcasinhl (C99)(C99)(C99) | computes the complex arc hyperbolic sine (function)
cacoshcacoshfcacoshl (C99)(C99)(C99) | computes the complex arc hyperbolic cosine (function)
ctanhctanhfctanhl (C99)(C99)(C99) | computes the complex hyperbolic tangent (function)
atanhatanhfatanhl (C99)(C99)(C99) | computes inverse hyperbolic tangent (\({\small\operatorname{artanh}{x} }\)artanh(x)) (function)
C++ documentation for atanh

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